Surface Dirac cones in three-dimensional topological insulators have generated tremendous and enduring interest for almost two decades owing to hosting a multitude of exotic properties. In this work, we unveil the existence of two types of anomalous surface Dirac cones in three-dimensional Dirac semimetals. These surface Dirac cones are located at the surfaces perpendicular to the rotation symmetry axis and are found to display a number of features remarkably different from that in topological insulators. The most prominent one is the absence of a singular Dirac node. In addition, the spin textures of these nodeless surface Dirac cones are found to exhibit a unique two-phase-angle dependence, leading to the presence of two different winding numbers in the orbital-resolved spin textures, which is rather different from the well-known spin-momentum locking in topological insulators. Despite the absence of a Dirac node, we find that the two types of surface Dirac cones are also characterized by quantized $\pi$ Berry phases, even though one of them takes a quadratic dispersion. In the presence of time-reversal-symmetry-breaking fields, we find that the responses of the surface and bulk Dirac cones display an interesting bulk-surface correspondence. The uncovering of these nodeless surface Dirac cones broadens our understanding of the topological surface states and bulk-boundary correspondence in Dirac semimetals and also lays down the basis for studying unconventional Dirac physics.

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• Topological materials
• Dirac semimetal
• Discrete symmetries in condensed matter
• Tight-binding model
• Trial wave functions
• k dot p method